Simulating the Ising Model: Algorithms, Code, and Results

Reference Python code (Metropolis and Wolff)

python
 ffON2NH02oMAcqyoh2UU MQCbz04ET5EljRmK3YpQ CPXAhl7VTkj2dHDyAYAf” data-copycode=“true” role=“button” aria-label=“Copy Code”>Copy CodeCopied
# metropolis_vs_wolff.py
import numpy as np import random from collections import deque 
def neighbors(L):
# return neighbor index deltas for 2D square lattice
    return [(-1,0),(1,0),(0,-1),(0,1)]

def energy(lattice, J=1.0):
    L = lattice.shape[0]
    E = 0
    for i in range(L):
        for j in range(L):
            s = lattice[i,j]
            E -= J  s  (lattice[(i+1)%L,j] + lattice[i,(j+1)%L])
    return E 
def magnetization(lattice):
    return lattice.sum()

def metropolis_step(lattice, T, J=1.0):
    L = lattice.shape[0]
    for _ in range(LL):
        i = np.random.randint(0,L); j = np.random.randint(0,L)
        s = lattice[i,j]
        nb = lattice[(i+1)%L,j] + lattice[(i-1)%L,j] + lattice[i,(j+1)%L] + lattice[i,(j-1)%L]
        dE = 2  J  s  nb         if dE <= 0 or np.random.rand() < np.exp(-dE/T):
            lattice[i,j] = -s 
def wolff_step(lattice, T, J=1.0):
    L = lattice.shape[0]
    p_add = 1 - np.exp(-2J/T)
    i0 = np.random.randint(0,L); j0 = np.random.randint(0,L)
    cluster_spin = lattice[i0,j0]
    stack = [(i0,j0)]
    cluster = set([(i0,j0)])
    while stack:
        i,j = stack.pop()
        for di,dj in [(-1,0),(1,0),(0,-1),(0,1)]:
            ni, nj = (i+di) % L, (j+dj) % L             if (ni,nj) not in cluster and lattice[ni,nj] == cluster_spin:
                if np.random.rand() < p_add:
                    cluster.add((ni,nj))
                    stack.append((ni,nj))
    for (i,j) in cluster:
        lattice[i,j] = -lattice[i,j]
    return len(cluster)

# Example run
def run_simulation(L=32, T=2.269, steps=20000, equil=5000, algo=‘metropolis’):
    lattice = np.random.choice([1,-1], size=(L,L))
    E_hist = []; M_hist = []
    for t in range(steps):
        if algo==‘metropolis’:
            metropolis_step(lattice, T)
        else:
            wolff_step(lattice, T)
        if t >= equil:
            E_hist.append(energy(lattice))
            M_hist.append(abs(magnetization(lattice)))
    E_avg = np.mean(E_hist); M_avg = np.mean(M_hist)
    Cv = (np.var(E_hist)) / (TT) / (LL)
    Chi = (np.var(M_hist)) / T / (LL)
    return {‘E’:E_avg/(LL), ’M’:M_avg/(LL), ‘Cv’:Cv, ‘Chi’:Chi}

if name == ‘main’:
    res = run_simulation(L=32, T=2.269, steps=20000, equil=5000, algo=‘wolff’)
    print(res)

Suggested experiments and results to produce

• Temperature sweep: simulate a range T ∈ [1.0, 3.5] with fine spacing near Tc and plot ⟨|M|⟩, Cv, χ versus T for several L (e.g., L=16,32,64).
• Finite-size scaling: locate pseudo-critical temperatures Tc(L) from peaks of Cv or χ and extrapolate Tc(L -> ∞).
• Autocorrelation: measure integrated autocorrelation times τint for energy and magnetization for Metropolis vs Wolff near Tc.
• Performance: compare wall-clock time per independent sample for Metropolis and Wolff; show Wolff produces decorrelated samples far faster near criticality.

Typical expected findings

• 2D Ising shows a second-order phase transition at Tc ≈ 2.269 (J/kB = 1 units).
• Metropolis exhibits critical slowing down (large τint near Tc); cluster algorithms (Wolff/Swendsen–Wang) mitigate this effectively.
• Specific heat shows a peak that sharpens with L; magnetization drops rapidly around Tc.
• Finite-size scaling collapses with known critical exponents (β=⁄₈, ν=1).

Tips for robust results

• Use many independent runs and compute error bars (bootstrap or jackknife).
• Ensure equilibration is long enough; verify by starting from cold and hot initial states and checking convergence.
• Use improved estimators where available (e.g., cluster algorithms give better variance properties).
• Store random seeds for reproducibility.

Extensions

• Add external magnetic field h: H = −J Σ⟨i,j⟩ s_i s_j − h Σ_i s_i to study hysteresis.
• Simulate 3D lattice (no exact solution; larger critical exponents and higher Tc).
• Study diluted Ising models, Potts models, or quantum Ising variants (requires different algorithms).

Conclusion

Combining Metropolis (or Glauber) for general-purpose sampling and Wolff/Swendsen–Wang for critical-region efficiency gives a practical and powerful simulation toolkit. Systematic temperature sweeps, finite-size scaling, and careful statistical analysis yield accurate characterization of phase behavior and critical exponents.